# Spin, Orbital, Weyl and Other Glasses in Topological Superfluids

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## Abstract

One of the most spectacular discoveries made in superfluid \(^3\)He confined in a nanostructured material like aerogel or nafen was the observation of the destruction of the long-range orientational order by a weak random anisotropy. The quenched random anisotropy provided by the confining material strands produces several different glass states resolved in NMR experiments in the chiral superfluid \(^3\)He-A and in the time-reversal-invariant polar phase. The smooth textures of spin and orbital order parameters in these glasses can be characterized in terms of the randomly distributed topological charges, which describe skyrmions, spin vortices and hopfions. In addition, in these skyrmion glasses the momentum-space topological invariants are randomly distributed in space. The Chern mosaic, Weyl glass, torsion glass and other exotic topological states are examples of close connections between the real-space and momentum-space topologies in superfluid \(^3\)He phases in aerogel.

## Keywords

Topological matter Disorder Glass Superfluid \(^3\)He Larkin–Imry–Ma effect Skyrmion Hedgehog Hopfion Half-quantum vortex Chern mosaic## 1 Introduction

The spin-triplet *p*-wave superfluid phases of liquid \(^3\)He [1] immersed in the aerogel matrix provide the arena for experimental and theoretical investigations of different types of spin and orbital orientational disorder, induced by the quenched orientational disorder of the aerogel strands. Especially interesting phenomena are realized in the chiral superfluid \(^3\)He-A phase, which in addition to superfluidity has the signatures of the spin nematic [2, 3] and orbital ferromagnet. One of the most spectacular discoveries was the observation of the destruction of the long-range orientational order in \(^3\)He-A by a weak random anisotropy [4]—the so-called Larkin–Imry–Ma (LIM) effect [5, 6, 7] (see also review paper [8]). This is the orbital glass state of the chiral superfluid \(^3\)He-A—the bulk 3D topological system with smooth disorder in the field of the orbital vector \({\hat{\mathbf{l}}}\), which describes the orientation of the orbital magnetization of Cooper pairs in this chiral liquid. The smooth texture of the \({\hat{\mathbf{l}}}\)-vector can be characterized by the integer valued topological charges valid for the soft topological objects, such as 2D and 3D skyrmions [9], merons, continuous vortices [10], topological solitons, domain walls, monopoles and hedgehogs. Following the notations of Refs. [11, 12], the LIM orbital glass state can be called the *intrinsic orbital skyrmion glass*.

The intrinsic orbital glass state persists even when the global anisotropy of the aerogel strands is present. In the polar-distorted \(^3\)He-A phase (PdA phase), which is formed in the aerogel with “nematically ordered” strands, provided by the commercially available nafen material [13], the 2D LIM state is observed with the disordered planar texture of the \({\hat{\mathbf{l}}}\)-vector [14].

The intrinsic orbital glass is realized as an equilibrium state. Whether it is the true glass state or the orbital liquid is an open question. The \(^3\)He-A in aerogel may have many degenerate ground states (or nearly degenerate states) with the rare events of the transitions between the states. All these states have smaller energy compared to the ordered state of the orbital ferromagnet, and thus are not able to relax to the ordered state with long-range order.

In addition, there are the topological excited states on the background of the intrinsic LIM glass. In particular, it was found that the aerogel strands strongly pin the singular topological defects (with hard cores), such as quantized vortices and half-quantum vortices—Alice strings [15]. The disordered state which contains pinned vortices is obtained by the Kibble–Zurek mechanism: vortex nucleation by fast cooling through \(T_\mathrm{c}\) [16, 17, 18].

*vortex glass*. This vortex glass is very different from the Larkin vortex glass in superconductors, where vortices have preferred orientation of magnetic fluxes along the magnetic field. In the isotropic aerogel, vortices have random orientation of vortex lines. In the aerogel with preferable orientations of the strands, the vortices form the disordered Ising glass with the random distributions of the winding numbers \(N=+1\) and \(N=-1\), or \(N=+1/2\) and \(N=-1/2\) in case of half-quantum vortices in Fig. 1.

The aerogel and nafen also pin the topological defects with the cores of intermediate sizes, such as spin vortices in Fig. 1. All these possibilities, in addition to the fully equilibrium LIM state, give rise to a zoo of quasiequilibrium glass states with different types of the pinned topological excitations. These states can be obtained using different protocols, see Fig. 2.

The rich glass states in superfluid phases of \(^3\)He could be useful for studies of different problems related to spin glasses [19, 20, 21]. Our measurements in superfluid \(^3\)He confined within nafen, Fig. 3, demonstrate that there are at least three types of spin-glass states with different NMR signatures. One of them is the equilibrium spin-glass state: due to spin–orbit interaction the orbital glass serves as the quenched orientational disorder acting on the spin-nematic vector \({\hat{\mathbf{d}}}\). The \(\hat{\mathbf {d}}\) vector is a unit vector (director) along the spontaneous uniaxial spin anisotropy of the A phase. As a result the LIM state of the \({\hat{\mathbf{d}}}\) vector is formed with the characteristic LIM scale larger than LIM scale in the orbital glass.

*spin-skyrmion glass*.

The *spin-vortex glass* states are obtained by cooling from the normal liquid to the polar phase under strong magnetic perturbations. This spin glass can be represented as a chaotic system of spin vortices pinned by the aerogel.

Here, we consider how different topological charges characterize different types of glass states and apply the simplest Larkin–Imry–Ma arguments to describe the properties of these glasses leaving a more detailed consideration for the future.

Topology of skyrmion glasses is discussed in Sects. 2 and 3; the orbital LIM glass is in Sect. 5; the spin glasses including spin-vortex glass are in Sect. 6; the vortex glasses are in Sect. 7: the combined effect of real and momentum-space topology is in Sect. 8.

## 2 2D Skyrmion Glass

### 2.1 Skyrmionic Topology of 2D Glass

### 2.2 Fluctuations of Topological Charges in 2D Skyrmion Glass

*L*inside the glass, which is much smaller than the dimension of the system and much larger than the LIM scale \(\xi _{\mathrm{LIM}}\). Then, the total charge \(Q_2\) in this region is fluctuating. In general, one may expect

*S*of the region, and one has \(m=2\):

## 3 3D Skyrmion Glasses

### 3.1 Topology of 3D Skyrmion Glasses

### 3.2 Fluctuations of Hopf Topological Charge in 3D Skyrmion Glass

*V*which is much smaller than the total volume of the sample and much larger than the volume of LIM scale. If the regions with positive and negative \(q_3(x)\) are randomly distributed, and we assume the Gaussian distribution, then \(\langle Q_3^2\rangle \) is proportional to the volume

*V*:

### 3.3 Fluctuations of \(Q_2\) Topological Charge in 3D Skyrmion Glass

## 4 Glasses in Chiral Superfluids

*p*-wave order parameter \({\varDelta }_{\alpha \beta }({\mathbf {k}}) = (i\sigma ^2 \sigma ^{\mu })_{\alpha \beta }k^iA_{\mu i}\) of chiral superfluid \(^3\)He-A is given by

^{1}The unit vector \({\hat{\mathbf{l}}}={\hat{\mathbf{e}}}_1\times {\hat{\mathbf{e}}}_2\) plays several roles in chiral superfluid: it shows the direction of the orbital angular momentum of Cooper pairs and thus determines the orbital magnetization of chiral superfluid; it determines the easy axis of the orbital anisotropy; it shows the direction to the Weyl nodes in the fermionic quasiparticle spectrum in momentum space; together with vectors \({\hat{\mathbf{e}}}_1\) and \({\hat{\mathbf{e}}}_2\) it forms the analog of the tetrad fields in general relativity; it is also responsible for continuous vorticity of the superflow velocity.

*U*(1) gauge field in Eq. (6):

- (i)
First is the equilibrium orbital LIM glass state of the orbital vector \({\hat{\mathbf{l}}}\) in Sect. 5. This equilibrium LIM state is obtained by slow cooling from the equilibrium normal (paramagnetic) state through the superfluid transition temperature \(T_\mathrm{c}\) [4].

- (ii)
Due to a weak spin–orbit interaction, the obtained random orientation of the orbital vector \({\hat{\mathbf{l}}}\) in turn serves as the quenched random anisotropy disorder for the spin-nematic vector \({\hat{\mathbf{d}}}\). As a result, the equilibrium spin-nematic LIM glass state is formed, with much larger length scale, \(\xi _{\mathrm{LIM}d} \gg \xi _D \gg \xi _{\mathrm{LIM}l}\), where \(\xi _D\) is the characteristic length scale of spin–orbit interaction, see Sect. 6.1.

- (iii)
There is also the nonequilibrium spin-nematic skyrmion glass state. It is obtained when the large enough resonant continuous radio-frequency excitation is applied during the cooling through \(T_\mathrm{c}\) [4]. The characteristic length scale of this \({\hat{\mathbf{d}}}\) glass is smaller than \(\xi _D\), see Sect. 6.2. In theory such metastable skyrmion glass is obtained by relaxation from the random initial configurations of the order parameter [34].

## 5 Orbital LIM Glass

### 5.1 Larkin–Imry–Ma Orbital Glass in Isotropic Aerogel

*V*is \(|N_{\mathrm{Hopf}}| \sim (V/\xi _{\mathrm{LIM}}^3)^{1/2}\).

### 5.2 LIM Glass in Anisotropic Aerogel

### 5.3 Superfluidity of 3D Skyrmion Glass in \(^3\)He-A

There were several suggestions that in chiral superfluids, superfluidity can be destroyed by skyrmions [7, 43].

*L*behavior of the exponent of the loop function means that superfluidity is not destroyed by the LIM texture. The nonzero superfluid density of the LIM state in \(^3\)He-A has been measured [45, 46, 47]. That is why the LIM state in \(^3\)He-A represents the system where the off-diagonal long-range order is destroyed. This is a 3D analog of the 2D Berezinskii–Kosterlitz–Thouless superfluid state. \(^3\)He-A represents the

*amorphous topological superfluid*. The nonzero superfluid density \(\rho _s\) means that the coarse grained

*U*(1) gauge field has a mass. Such glass state with nonzero mass of the effective gauge field corresponds to the confined phase suggested in Ref. [48].

The statement in Ref. [7] has been based on assumption of the \(m=2\) scaling law, which is not correct. In case of LIM state, the superfluidity is preserved due to \(m=1\) scaling. But the LIM state can be considered as a heat-insulator phase, since the lowest-energy fermionic states, which live near the Weyl nodes, can be localized. In principle, one may construct (possibly nonequilibrium) states with orbital disorder, in which the mass (charge) superfluidity is lost. Other states are possible when the mass superfluidity is lost, but the spin superfluidity retains, or vice versa: the spin superfluidity is lost, but the mass superfluidity is not, see Sect. 6.3. Such states would provide an analog of the separate charge and spin localization under random field [49]. However, it is not excluded that whatever is the scaling law, the glass state remains superfluid because of the pinning of the texture.

### 5.4 Skyrme Superfluid Versus Skyrme Insulator

Another theoretical challenge is the stability of superflow in pure \(^3\)He-A. It has been suggested that easy creation of skyrmions by the mass current destroys the superfluidity, and possible corresponding nonsuperfluid state has been called *Skyrme insulator* [43]. In reality, however, a finite-size system remains superfluid since the Feynman critical velocity \(v_{\mathrm{Feynman}}\), approximately inversely proportional to the system size, is not zero. In a channel of finite thickness, both the creation of skyrmion in \(^3\)He-A and creation of vortex ring in superfluid \(^4\)He require the overcoming of the critical velocity, at which the creation of these objects become energetically favorable. The Feynman critical velocity for creation of a vortex ring in superfluid \(^4\)He is \(v_{\mathrm{Feynman}} \sim (\hbar / md) \ln (d/a)\), where *d* is the width of the channel or slab, and *a* is the core size of the vortex. For skyrmions, the core size \(a\sim d\), and \(v_{\mathrm{Feynman}} \sim \hbar / md\), which is only logarithmically smaller than in superfluid \(^4\)He.

The instability of the supercurrent toward creation of skyrmions has been measured in \(^3\)He-A, see discussion in Ref. [40]. The measured threshold is much larger than the Feynman critical velocity. The reason is that while the creation of the skyrmions is energetically becomes favorable, the superflow is locally stable and the potential barrier for creation is by many orders of magnitude larger than the temperature of the system. That is why the skyrmions are created at the critical velocity, at which the helical instability of the orbital texture develops. In principle, one can construct geometry in which the superflow is locally unstable. In this case, the critical velocity will be reduced to the Feynman critical velocity.

## 6 Spin Glasses

### 6.1 Cascade of LIM Processes

*cascade LIM processes*: the quenched orientational disorder of aerogel strands on nanoscales gives rise to the orientational disorder in the orbital vector field (

*orbital glass state*) on a microscale, which in turn leads to the spin disorder (

*spin-glass state*) on a milliscale. According to the NMR measurements [4], the LIM scale for the disorder of the orbital vector \({\hat{\mathbf{l}}}\) is smaller than the characteristic scale of spin–orbit interaction, \(\xi _{\mathrm {LIM}l} \ll \xi _D\). Then, the corresponding LIM scale of the disordered state of the spin-nematics vector \({\hat{\mathbf{d}}}\) is:

*spin-glass state*can be characterized by its own \(\pi _2(S^2)\) and \(\pi _3(S^2)\) topological numbers. The latter is the spin Hopf invariant:

*orbital glass*and

*spin glass*represents the

*hierarchical double Skyrme glass*.

### 6.2 Skyrmion Spin Glass

The nonequilibrium skyrmion glass state originally has been obtained when the large enough resonant continuous radio-frequency excitation has been applied during the cooling through \(T_\mathrm{c}\) from the normal state to \(^3\)He-A [4]. The NMR signature of this state demonstrates that the characteristic length scale of textures in this \({\hat{\mathbf{d}}}\)-glass is smaller than \(\xi _D\), contrary to the equilibrium spin glass in Eq. (21). The nonequilibrium skyrmion glass with the same NMR signature can be also obtained by warming from the B phase to the A phase through the first-order phase transition, see Fig. 2 and spectra *(3)* and *(4)* in Fig. 5. Since such spin glass exists due to spin–orbit interaction with orbital spin glass, which disappears on transition from the A phase to the polar phase, the spin-skyrmion glass is annealed on this transition. On return back from the polar phase to the A phase, we observe change from spectra *(3)* and *(4)* to *(1)* and *(2)*, respectively.

### 6.3 Spin-Vortex Glass and Spin-Current Confinement

*(2)*and

*(4)*in Fig. 5.

*U*(1) spin vortices with density of topological charge \(q_1\). This is similar to the effective gauge field representing the equivalent description of disorder in terms the distributed linear topological defects in, e.g., Refs. [20, 50], where in particular the spin glass has been treated in terms of the effective

*SU*(2) gauge field. The noise in the distribution of positively charged \(Q_1=+1\) and negatively charged \(Q_1=-1\) spin vortices gives:

## 7 Vortex Glasses

### 7.1 Glass of Half-Quantum Vortices

The observed vortex glass—the *Alice glass*—differs from the Larkin vortex glass in superconductors, where vortices have preferred orientation of magnetic fluxes along the magnetic field. In the isotropic aerogel vortices have random orientation of vortex lines. In the aerogel with preferable orientations of the strands, the pinned vortex segments tend to align along the stands and thus they form disordered Ising glass with the random distributions of the winding numbers \(N = +1/2\) and \(N = -1/2\). This configuration is, however, not uniform along the strand direction, since the same vortex can be pinned by different strands in different parts of the sample. This type of vortex matter adds to the Zoo of vortex states in superconductors: Bragg glass, vortex glass, vortex liquid and the Abrikosov lattice [56, 57].

Several types of the solitonic glass are possible. Spin solitons are formed in the glass or the lattice of half-quantum vortices. They are formed between the half-quantum vortices due to spin–orbit interaction. In the vortex glass, they form the solitonic glass, and in the vortex lattice—the analog of Bragg glass (*solitonic Bragg glass*).

## 8 Discussion: Topological Fermionic Glasses

The spin-triplet superfluid phases of \(^3\)He have rich topological properties, which allow us to produce many types of the glass states classified in terms of the pinned topological defects (Alice strings, monopoles, domain walls, etc.) and textures (skyrmions, hopfions, merons, solitons, etc.). Some of these states have been experimentally identified in NMR experiments, but many other states are still waiting for their strong identification. Experimental and theoretical study of these states may lead to discovery of new phenomena and new concepts in the physics of the topological disorder.

- 1.
The A phase and the polar-distorted A phase are Weyl superfluids with Weyl nodes in the fermionic spectrum. The Weyl points serve as the Berry phase magnetic monopoles, but now in momentum space [58]. The corresponding Hamiltonian for quasiparticles near the Weyl points has the form: \(H=e^i_a(p_i - qA_i)\sigma ^a\), where \(\sigma ^a\) are the Pauli matrices in the Bogoliubov–Nambu particle-hole space; \(e^i_a\) are the elements of the effective (synthetic) tetrad field; \(\mathbf{A}=k_F{\hat{\mathbf{l}}}\) is the effective (synthetic) electromagnetic field; and \(q=\pm 1\) is effective electric charge.

- 2.
The polar phase has Dirac nodal line in the fermionic spectrum and correspondingly the degenerate tetrad field [59].

- 3.
The B phase and the polar-distorted B phase are fully gapped topological superfluids of the DIII class with Majorana fermions on the surface. These phases become the higher-order topological superfluids in applied magnetic field, see, e.g., Refs. [60, 61]

In particular, in the disordered LIM state of \(^3\)He-A in isotropic aerogel, the positions \(\pm k_F{\hat{\mathbf{l}}}\) of the Weyl nodes and the orientations of the tetrads \(e^i_a\) are smoothly and randomly distributed in space forming a unique example of a *Weyl glass*. The random positions of the nodes give rise to the random effective gauge field \(\mathbf{A}=k_F\nabla \times {\hat{\mathbf{l}}}\), while the random orientations of the tetrads with \(\left\langle e_a^\mu \right\rangle =0\) form the analog of the *torsion foam* in quantum gravity [66, 67].

Smooth disorder of superfluid phases of \(^3\)He allows us to consider the disordered state as collection of domains with different values of the momentum-space invariants—the Chern numbers [64, 68]. Such topological glass state represents the real-space analog of the Chern mosaic in the space of parameters [69, 70].

## Footnotes

- 1.
Traditionally in \(^3\)He literature, the orbital vector \(\hat{\mathbf {e}}_1\) is named \(\hat{\mathbf {m}}\) and vector \(\hat{\mathbf {e}}_2\) is named \(\hat{\mathbf {n}}\). We use \(\hat{\mathbf {e}}_{1,2}\) notation to stress connection to tetrad field. Also the vector \(\hat{\mathbf {e}}\) introduced later in the polar phase is usually named \(\hat{\mathbf {m}}\).

## Notes

### Acknowledgements

We thank Vladimir Dmitriev, Alexei Yudin and Samuli Autti for useful discussions. This work has been supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No. 694248).

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